A Tauberian theorem for ideal statistical convergence
aa r X i v : . [ m a t h . F A ] A ug A TAUBERIAN THEOREM FOR IDEAL STATISTICALCONVERGENCE
MAREK BALCERZAK AND PAOLO LEONETTI
Abstract.
Given an ideal I on the positive integers, a real sequence ( x n ) issaid to be I -statistically convergent to ℓ provided that (cid:8) n ∈ N : n |{ k ≤ n : x k / ∈ U }| ≥ ε (cid:9) ∈ I for all neighborhoods U of ℓ and all ε > . First, we show that I -statisticalconvergence coincides with J -convergence, for some unique ideal J = J ( I ) .In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel[analytic, coanalytic, resp.].Then we prove, among others, that if I is the summable ideal { A ⊆ N : P a ∈ A /a < ∞} or the density zero ideal { A ⊆ N : lim n →∞ n | A ∩ [1 , n ] | = 0 } then I -statistical convergence coincides with statistical convergence. This canbe seen as a Tauberian theorem which extends a classical theorem of Fridy.Lastly, we show that this is never the case if I is maximal. Introduction
Let
I ⊆ P ( N ) be an ideal, that is, a collection of subsets of the positive integers N closed under taking finite unions and subsets. It is also assumed that I containsthe collection Fin of finite subsets of N and that, unless otherwise stated, I isproper, that is, it is different from P ( N ) . Among the most important ideals wecan find the family of asymptotic density zero sets Z := (cid:26) A ⊆ N : lim n →∞ | A ∩ [1 , n ] | n = 0 (cid:27) and the summable ideal I /n := ( S ⊆ N : X n ∈ S n < ∞ ) . Mathematics Subject Classification.
Primary: 40A35, 11B05. Secondary: 54A20.
Key words and phrases.
Ideal statistical convergence; Tauberian condition; submeasures;generalized density ideal; maximal ideals.P.L. is supported by the Austrian Science Fund (FWF), project F5512-N26.
Marek Balcerzak and
Paolo Leonetti
Let X be a Hausdorff topological space. Given an ideal I , a sequence ( x n ) taking values in X is said to be I -convergent to ℓ ∈ X , in short x n → I ℓ , if { n ∈ N : x n / ∈ U } ∈ I for all neighborhoods U of ℓ . In the literature, Z -convergence is usually called statistical convergence , see [2, 7] and references therein. We recall that, if I 6 = Fin and X has at least two distinct points, then I -convergence does not correspondto ordinary convergence with respect to any topology on the same base set, see[21, Example 2.2] and [19, Proposition 4.2]. In particular, the notion of idealconvergence is a “proper extension” of classical convergence.Recently, Das and Savas introduced in [6] the notion of I -statistical convergence :a sequence ( x n ) taking values in X is said to be I -convergent to ℓ ∈ X if (cid:26) n ∈ N : |{ k ∈ [1 , n ] : x k / ∈ U }| n ≥ ε (cid:27) ∈ I (1)for all neighborhoods U of ℓ and all ε > (note that the original definition hasbeen given in the context of normed spaces). The aim of this article is threefold.First, the authors of [6] remark that Fin -statistical convergence corresponds tostatistical convergence, cf. also [5, Remark 1]. Hence, one may wonder whether I -statistical convergence corresponds to J -convergence, for some ideal J = J ( I ) .We give a positive answer, in a slightly more general context, see Theorem 2.3.Second, the same authors claim in [6, Remark 2] that there exists a sequence ( x n ) which is Z -statistically convergent but not statistically convergent. However,it turns out that their claim is false. Indeed, we show that Z -statistical conver-gence and statistical convergence coincide, see Theorem 2.7 and Corollary 2.8. Aswe will explain in the next Section, this is a Tauberian theorem which extendsa classical result of Fridy [13, Theorem 3]. Related results have been extensivelystudied in the literature, see e.g. [3, 9, 14, 15, 16, 24, 25, 26, 27, 28].Lastly, on the opposite direction, we prove that I -statistical convergence never coincides with statistical convergence whenever I is maximal, see Theorem 2.10.2. Main results
An ideal I is said to be a P-ideal if it is σ -directed modulo finite sets, i.e., foreach sequence ( A n ) in I there exists A ∈ I such that A n \ A is finite for all n . Byidentifying sets of integers with their characteristic functions, we equip P ( N ) withthe Cantor-space topology and therefore we can assign the topological complexityto the ideals on N . In particular, we can speak about Borel ideals, analytic ideals,meager ideals, etc. It is a folklore result that the the ideals with lowest topologicalcomplexity are F σ -ideals. We refer to [18] for a recent survey on ideals and filters. Tauberian theorem for ideal statistical convergence
3A map ϕ : P ( N ) → [0 , ∞ ] is a submeasure provided that for all A, B ⊆ N :(i) ϕ ( ∅ ) = 0 , (ii) ϕ ( A ) ≤ ϕ ( B ) if A ⊆ B , (iii) ϕ ( A ∪ B ) ≤ ϕ ( A ) + ϕ ( B ) , and(iv) ϕ ( { n } ) < ∞ for all n . In addition, a submeasure ϕ is lower semicontinuousif: (v) ϕ ( A ) = lim n →∞ ϕ ( A ∩ [1 , n ]) for all A . By a classical result of Solecki, a(not necessarily proper) ideal I is an analytic P-ideal if and only if there existsa lower semicontinuous submeasure ϕ such that I coincides with the exhaustiveideal Exh( ϕ ) generated by ϕ , that is, I = Exh( ϕ ) := { A ⊆ N : lim n →∞ ϕ ( A \ [1 , n ]) = 0 } and ϕ ( N ) < ∞ , cf e.g. [10, Theorem 1.2.5]. Definition 2.1.
A sequence of submeasures µ = ( µ n ) is said to be smooth pro-vided that:( s
1) for all n ∈ N , µ n is supported on a nonempty set I n ;( s lim n →∞ µ n ( { k } ) = 0 for all k ∈ N ;( s lim sup n →∞ µ n ( N ) > .In this regard, let Z µ be the ideal defined by Z µ := (cid:26) A ⊆ N : lim sup n →∞ µ n ( A ∩ I n ) = 0 (cid:27) . Note that, if I is an ideal on N , then there exists a smooth sequence of sub-measures µ such that I = Z µ : indeed, it is sufficient to set µ n ( A ) equal to thecharacteristic function P ( N ) \I ( A ) for each A ⊆ N and n ∈ N .If, in addition, ( I n ) is a partition of N into finite nonempty sets then Z µ is a generalized density ideal , as introduced by Farah in [11, Section 2.10], cf. also[12]. Recall that every generalized density ideal is an analytic P-ideal: indeed, Z µ coincides with Exh( ϕ µ ) , where ϕ µ is the lower semicontinuous submeasure sup k µ k . The class of generalized density ideals is very rich, including for exampleall Erdős–Ulam ideals (among others, Z ), the Fubini product ∅ × Fin , simpledensity ideals [1], and ideals defined in [23] by Louveau and Veličković, cf. [20,Section 2] and references therein.
Definition 2.2.
Let I be an ideal and µ = ( µ n ) be a smooth sequence of sub-measures. A sequence ( x n ) taking values in a Hausdorff topological space X issaid to be ( I , µ ) -convergent to ℓ , shortened with x n → ( I , µ ) ℓ , if { n ∈ N : µ n ( { k ∈ N : x k / ∈ U } ) / ∈ V } ∈ I for each neighborhood U of ℓ ∈ X and V of ∈ R .In other words, the sequence ( x n ) is ( I , µ ) -convergent to ℓ if and only if µ n ( { k ∈ N : x k / ∈ U } ) → I Marek Balcerzak and
Paolo Leonetti for each neighborhood U of ℓ . Moreover, it is clear that x n → (Fin , µ ) ℓ if and onlyif x n → Z µ ℓ . This observation is generalized in Theorem 2.3 below.Hereafter, let λ = ( λ n ) be the sequence of uniform probability measures on N ∩ [1 , n ] , that is, λ n ( A ) = | A ∩ [1 , n ] | n (2)for all n ∈ N and A ⊆ N . Then it is easy to see that, for each ideal I , ( I , λ ) -convergence corresponds with I -statistical convergence defined in (1). Note that ( I , µ ) -convergence includes also the case of I -lacunary statistical convergencewhere each µ n is the uniform probability measure on N ∩ [ a n , a n +1 ) such that ( a n ) is an increasing sequence of positive integers for which a n +1 − a n → ∞ , cf.[5, Definition 6].We are ready to state our main results (all the proofs are given in Section 3).Let us start with an equivalence with the classical notion of ideal convergence. Theorem 2.3.
Let I be an ideal and µ be a smooth sequence of submeasures. Thenthere exists a unique ideal J = J ( I , µ ) such that ( I , µ ) -convergence coincides with J -convergence. In addition, J is proper if and only if µ n ( N ) I . To be precise, we say that ( I , µ ) -convergence "coincides" with J -convergenceif, for some Hausdorff space X with at least two points, every sequence ( x n ) takingvalues in X is ( I , µ ) -convergent to ℓ ∈ X if and only if ( x n ) is J -convergent to ℓ .At this point, one may ask whether there is some relationship between the pair ( I , µ ) and the ideal J ( I , µ ) in Theorem 2.3. First of all, we prove that, undersome mild conditions, J ( I , µ ) has the same topological complexity of I . Theorem 2.4.
Let I be an ideal and µ be a smooth sequence of lower semi-continuous submeasures. Then the ideal J ( I , µ ) is Borel [ analytic, coanalytic,respectively ] whenever I is Borel [ analytic, coanalytic, resp. ] . It turns out that if the pair ( I , µ ) is "sufficiently nice" then we can find theideal J ( I , µ ) explicitly. To this aim, we need the following definitions. Definition 2.5.
Given a real α > , we say that an ideal I is α -thick providedthat A / ∈ I whenever there exist a real c > and infinitely many n ∈ N such that N ∩ [ n, n + cn α ] ⊆ A .It turns out that Z is -thick, cf. the proof of Corollary 2.8. On the other hand,if I is maximal ideal, then I is not α -thick, for every α > . Indeed, exactly oneamong the sets A := N ∩ S n [ a n , a n +1 ] and A c belongs to I , where a := 1 and a n +1 := 2 a n for all n ∈ N , and such a set contains infinitely many intervals of thetype N ∩ [ n, n + cn α ] , for each c > . Tauberian theorem for ideal statistical convergence Definition 2.6.
Given a real α > , we say that a sequence of submeasures µ = ( µ n ) is α -flat provided that, for each A ⊆ N , there exists a real d = d ( A ) > such that | µ n +1 ( A ) − µ n ( A ) | ≤ d/n α for all n ∈ N .It is easy to show that the sequence λ defined in (2) is -flat, cf. the proofof Corollary 2.8 for details. More generally, for each α > , a family of α -flatsequences is given as follows. Suppose that µ is smooth and each µ n is a probabilitymeasure supported on I n := N ∩ [1 , ι n ] , where ( ι n ) is an increasing sequence in N such that:(i) ι n ≤ bn β for all n and some b, β > ;(ii) ι n +1 − ι n ≤ cn γ for all n and some c, γ > ;(iii) | µ n +1 ( { k } ) − µ n ( { k } ) | ≤ d/n δ for all n , all k ≤ ι n , and some d, δ > ;(iv) µ n ( { k } ) ≤ e/n η for all n , all k > ι n , and some e, η > ;(v) α ≤ min { δ − γ, η − β } .Then it is routine to check that µ is α -flat. Similarly, if we assume for simplicitythat ι n = n for all n , then Z µ is α -thick, for some α ∈ (0 , , provided that µ n ( N ∩ [ n − cn α , n ]) for all c > . It is worth to remark that, in such cases,the ideal Z µ corresponds to the ideal generated by the nonnegative regular matrix R = { r n,k : n, k ∈ N } , where r n,k := µ n ( { k } ) , cf. e.g. [4, Section 2].With these premises, we can state the following characterization. Theorem 2.7.
Let ν and µ be two smooth sequences of submeasures such that Z ν is α -thick and µ is α -flat, for some α ∈ (0 , . Then J ( Z ν , µ ) = Z µ , that is, ( Z ν , µ ) -convergence coincides with Z µ -convergence. Roughly, Theorem 2.7 states that if the pair ( I , µ ) is sufficiently nice, then µ n ( A ) → I implies µ n ( A ) → (3)for all A ⊆ N . At the point, if A is fixed, the real sequence ( x n ) defined by x n := µ n ( A ) is arbitrary, though nonnegative. Hence, in the case I = Z and | x n +1 − x n | ≤ d/n for all n and some d > (which corresponds to -flatness ofthe sequence µ relative to A ), the claim (3) can be rewritten as x n → Z implies x n → . Indeed, this is a classical result of Fridy, see [13, Theorem 3]. Here, he alsoproves that the Tauberian condition | x n +1 − x n | ≤ d/n is best possible. This hasbeen soon extended by Maddox, in the context of strong summability for slowlyoscillating sequences, see [24, 25]. A quite different Tauberian condition for Borel Marek Balcerzak and
Paolo Leonetti summability (related to / -flatness) can be found in [15]. Finally, there are relatedresults for statistically slowly oscillating sequences [3, 26, 27, 28] and for sequenceswhich satisfy a gap Tauberian condition [9, 16].As an application of Theorem 2.7, we obtain a sufficient condition for the equiv-alence between statistical convergence and I -statistical convergence. Corollary 2.8.
Let I be an ideal such that I ⊆ Z . Then I -statistical convergencecoincides with statistical convergence. Since Z is the ideal generated the upper asymptotic density d ⋆ defined by d ⋆ ( A ) := lim sup n →∞ | A ∩ [1 , n ] | n (4)for all A ⊆ N (that is, Z = { A ⊆ N : d ⋆ ( A ) = 0 } ), it follows that Corollary2.8 applies to all ideals I of the type { A ⊆ N : µ ⋆ ( A ) = 0 } , where µ ⋆ is an"upper density" on N , in the sense of [22], such that d ⋆ ≤ µ ⋆ pointwise. In par-ticular, possible choices for µ ⋆ are: the upper Banach density, the upper analyticdensity, the upper Pólya density, the upper Buck density, together with all upper α -densities with α ≥ (see [22] for details; cf. also [8] for the relationship betweenideals and "abstract densities").In addition, as a special instance of Corollary 2.8, we have: Corollary 2.9. I -statistical convergence coincides with statistical convergence if I = I /n or I = ∅ × Fin . Lastly, we show that the conclusion of Corollary 2.8 cannot be strenghtened tothe whole class of ideals I . Indeed, this is never the case if I is maximal. Theorem 2.10.
Let I be a maximal ideal. Then I -statistical convergence doesnot coincide with statistical convergence. To conclude, note that the definition of ( I , µ ) -convergence depends on the choiceof the sequence µ . Indeed, it is possible that ( I , µ ) -convergence does not coincidewith ( I , ν ) -convergence, where ν is another smooth sequence of submeasures forwhich Z µ = Z ν . Example 2.11.
Let x be the sequence defined by x n = A ( n ) for all n ∈ N ,where A := S n ≥ [ (2 n )! , (2 n + 1)! ] . Moreover, set µ n = λ n and ν n = d ⋆ , asdefined in (2) and (4), respectively, for all n ∈ N , so that Z µ = Z ν = Z . Notethat d ⋆ ( A ) = d ⋆ ( A c ) = 1 (in particular, x is not statistically convergent). Set A m := { n ∈ N : λ n ( A ) ≥ / m } for each m ∈ N . Then ( A m ) is an increasingsequence of sets and there exists a maximal ideal I containing all the A m ’s. It Tauberian theorem for ideal statistical convergence ε > , the set { n ∈ N : λ n ( n ) ≥ ε } is contained in some A m ∈ I , so that λ n ( A ) → I . Therefore x n → ( I , µ ) . On the other hand, { n ∈ N : ν n ( { k ∈ N : | x k − | ≥ / } ) ≥ / } = { n ∈ N : ν n ( { k ∈ N : x k = 0 } ) ≥ / } = { n ∈ N : d ⋆ ( A c ) ≥ / } = N / ∈ I , hence x n ( I , ν ) .We leave as open questions for the interested reader to "characterize" the class ofideals I for which I -statistical convergence coincides with statistical convergenceand to establish whether Theorem 2.10 holds for nonmeasurable ideals or thosewithout the Baire property. 3. Proofs
Before we start proving our results, we state the next lemma (which is straight-forward, we omit details):
Lemma 3.1.
Fix ideals I , J on N , let µ, ν be two smooth sequences of submea-sures, and fix α, β > . Then: (i) ( I , µ ) -convergence implies ( J , µ ) -convergence, provided that I ⊆ J ; (ii) ( I , µ ) -convergence implies ( I , ν ) -convergence, provided that, for all A ⊆ N , it holds ν n ( A ) ≤ µ n ( A ) for all sufficiently large n ; (iii) I is α -thick implies that J is α -thick, provided that J ⊆ I ; (iv) I is α -thick implies that I is β -thick, provided that α ≤ β ; (v) µ is α -flat implies that µ is β -flat, provided that β ≤ α . Thus, let us start with the proof of Theorem 2.3.
Proof of Theorem 2.3.
Given the pair ( I , µ ) , define the family J = J ( I , µ ) := { A ⊆ N : µ n ( A ) → I } . (5)It is clear that J is closed under subsets. Moreover, J is closed under finiteunions; indeed, for all A, B ∈ J and ε > , we have { n ∈ N : µ n ( A ) ≥ ε/ } ∈ I and { n ∈ N : µ n ( B ) ≥ ε/ } ∈ I ; hence { n : µ n ( A ∪ B ) ≥ ε } ⊆ { n : µ n ( A ) ≥ ε/ } ∪ { n : µ n ( B ) ≥ ε/ } ∈ I , so that A ∪ B ∈ J . Since µ is a smooth sequence of submeasures, we have lim n →∞ µ n ( { k } ) → for all k ∈ N , by ( s µ n ( { k } ) → I , whichimplies that Fin ⊆ J . This shows that J is an ideal on N and, in addition, J is proper if and only if µ n ( N ) I . At this point, let us prove that ( I , µ ) -convergence coincides with J -convergence. Let X be an Hausdorff space with Marek Balcerzak and
Paolo Leonetti at least two points, let us say a and b , and let ( x n ) be a sequence in X whichis ( I , µ ) -convergent to some ℓ ∈ X , that is, µ n ( { k ∈ N : x k / ∈ U } ) → I foreach neighborhood U of ℓ . By the definition of J in (5), this is equivalent to { k ∈ N : x k / ∈ U } ∈ J for each neighborhood U of ℓ , i.e., x n → J ℓ . Finally, letus suppose for the sake of contradiction that there exists another ideal J ′ = J such that ( I , µ ) -convergence coincides with J ′ -convergence. In particular, thereexists A ∈ J △J ′ and J -convergence coincides with J ′ -convergence. Let ( x n ) bethe sequence defined by x n = a if n ∈ A and x n = b otherwise. It follows thatexactly one of the conditions x n → J b and x n → J ′ b is true. This shows that J is unique, completing the proof. (cid:3) Note that the ideal J ( I , µ ) defined in (5) corresponds to Z µ if I = Fin . Proof of Theorem 2.4.
Let us rewrite the ideal J ( I , µ ) defined in (5) as follows: J ( I , µ ) = { A ⊆ N : ∀ m ∈ N , f m ( A ) ∈ I} , where, for each m ∈ N , f m : P ( N ) → P ( N ) is the function defined by f m ( A ) := { n ∈ N : µ n ( A ) > /m } for all A ⊆ N . At this point, by the lower semicontinuity of each µ n , we obtain f m ( A ) = S k ∈ N f m,k ( A ) , where, for each k ∈ N , f m,k : P ( N ) → P ( N ) is thefunction defined by f m,k ( A ) := { n ∈ N : µ n ( A ∩ [1 , k ]) > /m } for all A ⊆ N and m ∈ N . Therefore J ( I , µ ) = { A ⊆ N : ∀ m ∈ N , ∃ k ∈ N , f m,k ( A ) ∈ I} = \ m ∈ N [ k ∈ N f − m,k [ I ] . The claim follows by noting that f m,k is continuous and that the continuous preim-age of Borel [analytic, coanalytic, respectively] sets is Borel [analytic, coanalytic,resp.], cf. e.g. [29]. (cid:3) To prove the next result, we need the following intermediate lemma.
Lemma 3.2.
Let µ and ν be two smooth sequences of submeasures. Then Z µ -convergence implies ( Z ν , µ ) -convergence.Proof. Let ( x n ) be a sequence in a Hausdorff space X , fix ℓ ∈ X , and supposethat x n ( Z ν ,µ ) ℓ . Then there exist a neighborhood U of ℓ and a real ε > suchthat { n ∈ N : µ n ( { k ∈ N : x k / ∈ U } ) ≥ ε } / ∈ Z ν , that is, lim sup t →∞ ν t ( { n ∈ N : µ n ( { k ∈ N : x k / ∈ U } ) ≥ ε } ) > . Tauberian theorem for ideal statistical convergence ( t m ) and a real δ > such that ν t m ( { n ∈ I t m : µ n ( { k ∈ I n : x k / ∈ U } ) ≥ ε } ) ≥ δ for all m ∈ N . At this point, fix an integer n ∈ I t such that µ n ( { k ∈ I n : x k / ∈ U } ) ≥ ε and define recursively a sequence ( n h ) of positive integers as follows: foreach h ∈ N , let n h +1 be an integer greater than n h such that µ n h +1 ( { k ∈ I n h +1 : x k / ∈ U } ) ≥ ε ; note that such an integer exists because the sequence ( t m ) is infiniteand F := { n , . . . , n h } ∈ Fin ⊆ Z ν so that the sequence ν n ( F ) converges to and,in particular, it is smaller than δ whenever n is sufficiently large. It follows that µ n h ( { k ∈ I n h : x k / ∈ U } ) ≥ ε for all h ∈ N , which implies { k ∈ N : x k / ∈ U } / ∈ Z µ . Therefore ( x n ) Z µ ℓ . (cid:3) We proceed now to the proof of Theorem 2.7.
Proof of Theorem 2.7.
On the one hand, thanks to Lemma 3.2, Z µ -convergenceimplies ( Z ν , µ ) -convergence. Conversely, let ( x n ) be a sequence in a Hausdorffspace X which is not Z µ -convergent to ℓ ∈ X ; hence there exist a neighborhood U of ℓ , a real ε > , and an increasing sequence of positive integers ( n t ) such that µ n t ( { k ∈ N : x k / ∈ U } ) ≥ ε (6)for all t ∈ N . At this point, fix S ⊆ N . Since the sequence µ is α -flat, we obtainthat there exists d = d ( S ) > such that f S ( n ) ≤ d/n α for all n ∈ N , where f S ( n ) := | µ n ( S ) − µ n +1 ( S ) | for each n ∈ N . Claim Fix S ⊆ N . There exists a constant κ = κ ( α, d ( S )) > such that ⌊ cn α ⌋ X i =1 f S ( n + i ) ≤ κc. for all n ∈ N and reals c > . Proof.
First of all, for every n ∈ N and c > , we have the following upper bound ⌊ cn α ⌋ X i =1 f S ( n + i ) ≤ ⌊ cn α ⌋ X i =1 d ( n + i ) α ≤ Z cn α d ( n + t ) α d t. (7)0 Marek Balcerzak and
Paolo Leonetti
Hence, if α ∈ (0 , , we have ⌊ cn α ⌋ X i =1 f S ( n + i ) ≤ d − α (cid:0) ( n + cn α ) − α − n − α (cid:1) ≤ d − α n − α (cid:16)(cid:0) cn α − (cid:1) − α − (cid:17) ≤ d − α n − α · cn α − = cd − α . Similarly, if α = 1 , it follows by (7) that ⌊ cn α ⌋ X i =1 f S ( n + i ) ≤ d log (cid:18) n + cnn (cid:19) = d (log(1 + c )) ≤ cd, which completes the proof. (cid:3) Claim Fix S ⊆ N . Then for all δ > , there exist c > and n ∈ N suchthat | µ n + m ( S ) − µ n ( S ) | ≤ δ (8)for all integers n ≥ n and all integers m ∈ [0 , cn α ] . Proof.
With the same notation above, it follows by Claim 1 that for all n ∈ N ,all c > , and all integers m ∈ [1 , cn α ] , we have that | µ n + m ( S ) − µ n ( S ) | ≤ m X i =0 f S ( n + i ) ≤ f S ( n ) + κc ≤ dn α + κc. At this point, the wanted inequality (8) is obtained by choosing c > sufficientlysmall and n ∈ N sufficiently large so that d/n α + κc ≤ δ . (cid:3) To conclude, choosing δ = ε / and S = { k ∈ N : x k / ∈ U } in Claim 2 and usinginequality (6), we obtain that there exist c > and t ∈ N such that µ n t + m ( S ) ≥ µ n t ( S ) − ε / ≥ ε / for all integers t ≥ t and all integers m ∈ [ 0 , cn αt ] . Therefore A := { n ∈ N : µ n ( S ) ≥ ε / } ⊇ [ t ≥ t ( N ∩ [ n t , (1 + c ) n αt ]) . Since Z ν is α -thick, then A / ∈ Z ν . This implies that x n ( Z ν ,µ ) ℓ . (cid:3) Proof of Corollary 2.8.
Let λ be the smooth sequence of submeasures defined in(2). Thanks to Theorem 2.7, it is sufficient to show that Z = Z λ is -flat andthat I is -thick. To this aim, note that, for all A ⊆ N and n ∈ N , we have | λ n ( A ) − λ n +1 ( A ) | = (cid:12)(cid:12)(cid:12)(cid:12) λ n ( A ) − A ( n + 1) + nλ n ( A ) n + 1 (cid:12)(cid:12)(cid:12)(cid:12) ≤ n + 1 ≤ n . Tauberian theorem for ideal statistical convergence Z is -flat. Moreover, we claim that Z is -thick. Indeed, fix A ⊆ N suchthat there exist c > and infinitely many n ∈ N for which N ∩ [ n, (1 + c ) n ] ⊆ A .Then the upper asymptotic density of A is at least c / c > . Hence A / ∈ Z .Therefore I is -thick by Lemma 3.1(iii). (cid:3) Proof of Corollary 2.9.
It is known I /n ⊆ Z , hence the claim follows by Corollary2.8. Similarly, it is sufficient to prove that there exists a copy of the Fubini product I = ∅ × Fin on N which is contained in Z . Thus, note that I can be written as { A ⊆ N : ∀ k ∈ N , { n ∈ A : υ ( n ) = k − } ∈ Fin } , where υ ( n ) is the biggest exponent m ∈ N such that m divides n . Fix A ∈ I .Then, for each k ∈ N , there exists a finite set F = F ( k ) ⊆ N such that A \ F contains only multiples of k , so that the upper asymptotic density of A is most / k . By the arbitrariness of k , we conclude that A ∈ Z . Therefore I ⊆ Z . (cid:3) Proof of Theorem 2.10.
Let us suppose that I is a maximal ideal on N . Thanks toTheorem 2.3, there exists a unique ideal J such that ( I , λ ) -convergence coincideswith J -convergence; in addition, J := { A ⊆ N : λ n ( A ) → I } , see (5). Toconclude, we show that J 6 = Z by proving that they have different topologicalcomplexities. To this aim, it is sufficient to prove that J is nonmeasurable (onthe other hand, it is known that Z is a F σδ -ideal).Identifying each set A ⊆ N with the sequence ( A ( n ) : n ∈ N ) ∈ { , } N , wecan rewrite J as J = (cid:8) x ∈ { , } N : n P i ≤ n x i → I (cid:9) . Claim There exists an increasing sequence ( a n ) in N such that a n /a n +1 → as n → ∞ and S n ∈ N A n −
6∈ I , where A n := N ∩ ( a n − , a n ] and a := 0 . Proof.
Fix an increasing sequence ( a n ) in N such that a n /a n +1 → and, for each i ∈ { , , } , define R i := S n ≡ i mod 3 A n . Since I is maximal, there exists a unique i ⋆ ∈ { , , } such that R i ⋆ / ∈ I . If i ⋆ = 2 we are done. If i ⋆ = 0 [ i ⋆ = 1 ,respectively], just delete one element [two elements, resp.] from the sequence. (cid:3) At this point, fix an increasing sequence ( a n ) as in Claim 3 and define thefunction Λ : { , } N → { , } N by Λ( x ) = ( x a n − : n ∈ N ) for each sequence { , } N . Moreover, define the function f : N → N such that f ( n ) = k whenever n ∈ A k , and set F := { x ∈ { , } N : x i = x j whenever f ( i ) = f ( j ) , and x a n − = x a n − and x a n = 0 for all n ∈ N } , Marek Balcerzak and
Paolo Leonetti that is, F is the sequence of { , } -valued sequences which are constant on allintervals A n and A n − ∪ A n − , with value on all the A n ’s. Then the restrictionof Λ on F is an homeomorphism F → { , } N . Since F is closed, it is sufficientto show that Λ( J ∩ F ) is not measurable. Claim lim n →∞ (cid:16) x a n − a n P i ≤ a n x i (cid:17) = 0 for all x ∈ F . Proof.
Considering that x i = x j whenever f ( i ) = f ( j ) and that a n /a n +1 → as n → ∞ , we have that a n X i ≤ a n x i = 1 a n X i ≤ n x a i | A i | = (cid:18) − a n − a n (cid:19) x a n + 1 a n X i ≤ n − x a i | A i | . Therefore (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x a n − a n X i ≤ a n x i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n − a n x a n − a n X i ≤ n − x a i | A i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ a n − a n + 1 a n X i ≤ n − | A i | = 2 a n − a n → , which completes the proof. (cid:3) Lastly, we define the function g : N → N by g ( n ) := ⌈ n ⌉ for all n ∈ N (so that g ( f ( n )) = k if n ∈ A k − ∪ A k − ∪ A k ), and set S := { x ∈ { , } N : x n → U } , where U := { f − [ g − [ A ]] : A ∈ I} (note that U is a maximal ideal on N ). Claim Λ( J ∩ F ) = S . Proof.
First, we show that
S ⊆ Λ( J ∩ F ) . Fix a sequence y ∈ S and set U := { n ∈ N : y n = 1 } . We need to prove that there exists x ∈ J ∩ F such that Λ( x ) = y . Note that there is exactly one sequence x ∈ F such that Λ( x ) = y ,that is, the unique one in F such that x a n − = y n for all n ∈ N . Let us show that x ∈ J , i.e., n P i ≤ n x i → I . Since y ∈ S , then U ∈ U , hence ˜ U := f − [ g − [ U ]] = S u ∈ U ( A u ∪ A u − ∪ A u − ) ∈ I . This implies, thanks to Claim 4 (see also Figure 1 below), that, for each ε > ,there exists a finite set F ε ∈ Fin such that (cid:8) n ∈ N : n P i ≤ n x i ≥ ε (cid:9) ⊆ F ε ∪ ˜ U ∈ I . Therefore x ∈ J ∩ F and Λ( x ) = y . Tauberian theorem for ideal statistical convergence Λ( J ∩ F ) ⊆ S . Fix a sequence x ∈ J ∩ F .Then we need to show that Λ( x ) ∈ S , that is, x a n − → U . This is equivalent to V := { n ∈ N : x a n − = 1 } ∈ U and also, by the definition of U , to ˜ V := f − [ g − [ V ]] = S v ∈ V ( A v ∪ A v − ∪ A v − ) ∈ I . By assumption we know n P i ≤ n x i → I and the sequence x is constant on eachinterval A n , with x a n = 0 and x a n − = x a n − for all n . Fix a sufficiently small ε > , hence M := (cid:8) n ∈ N : n P i ≤ n x i ≥ ε (cid:9) ∈ I . Thanks to Claim 4, we obtain G := (cid:0)S v ∈ V A v − (cid:1) \ M ∈ Fin . Therefore, thanks to Claim 3, we conclude that ˜ V ⊆ (cid:16)S n A n (cid:17) ∪ S v ∈ V A v − ⊆ (cid:16)S n A n (cid:17) ∪ M ∪ G ∈ I , which completes the proof. (cid:3) Identifying { , } with the additive group Z , we have that S is a subgroup ofthe compact group { , } N . Moreover, S is not closed (since { ,...,k } ( n ) → U forall k ∈ N but N ( n ) U ) and it has finite index (since U is a maximal ideal,then there are exactly two cosets of S ). It follows by [17, Proposition 1.1(c)] that S is nonmeasurable. Thus, thanks to Claim 5, Λ( J ∩ F ) is nonmeasurable. (cid:3) a u − a u − a u − a u ε − ε • • • • n n ∈ A u − ∪ A u − ∪ A u : n P i ≤ n x i ≥ ε o Figure 1.
Graph of the sequence (cid:0) n P i ≤ n x i (cid:1) in the case x a u − = 1 . References
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